Zeros of one-forms and the topology of algebraic maps

Abstract

We construct a smooth complex projective variety whose Albanese morphism is a homotopy fiber bundle but not a submersion. The same variety fibers smoothly over the circle, although every holomorphic one-form on it has a zero. A second construction yields smooth complex projective varieties $X$ such that the Aomoto complex of every nonzero holomorphic one-form on every connected finite \'etale cover of $X$ is exact, while $X$ admits no real closed one-form without zeros. The two constructions build, respectively, on a homology fiber bundle of Corr\^ea--Koll\'ar that is not a homotopy fiber bundle and on a rational cohomology torus constructed by Debarre--Jiang--Lahoz. Consequently, we disprove Kotschick's conjecture, the remaining implication in the Bobadilla--Koll\'ar conjecture, and a conjecture of the first-named author.